The Black-Scholes Formula

• The Black-Scholes formula is a limiting case of the binomial formula (infinitely many periods) for the price of a European option.

• Consider an European call (or put) option written on a stock

• Assume that the stock pays dividend at the continuous rate \(\delta\)

Black-Scholes Formula (cont’d)

• Call Option price

\[ C(S, K, \sigma, r, T, \delta) = Se^{-\delta T} N(d_{1}) - Ke^{-r T} N(d_{2}) \]

• Put Option price

\[ P(S, K, \sigma, r, T, \delta) = Ke^{-r T} N(-d_{2}) - Se^{-\delta T} N(-d_{1}) \]

where

\[ \begin{aligned} d_{1} & = \frac{\ln(S / K) + (r - \delta + \frac{1}{2} \sigma^{2}) T}{\sigma \sqrt{T}} \\ d_{2} & = d_{1} - \sigma\sqrt{T} \end{aligned} \]

Black-Scholes Assumptions

• Assumptions about stock return distribution

• Continuously compounded returns on the stock are normally distributed and independent over time (no “jumps”)

• The volatility of continuously compounded returns is known and constant

• Future dividends are known, either as dollar amount or as a fixed dividend yield

Black-Scholes Assumptions (cont’d)

• Assumptions about the economic environment

• The risk-free rate is known and constant

• There are no transaction costs or taxes

• It is possible to short-sell costlessly and to borrow at the risk-free rate

Option Greeks

• What happens to the option price when one and only one input changes?

• Delta ( \(\Delta\) ): change in option price when stock price increases by \(\$ 1\)

• Gamma ( \(\Gamma\) ): change in delta when option price increases by \(\$ 1\)

• Vega: change in option price when volatility increases by \(1 \%\)

• Theta \((\theta)\) : change in option price when time to maturity decreases by 1 day

• Rho \((\rho)\) : change in option price when interest rate increases by \(1 \%\)

• Greek measures for portfolios

• The Greek measure of a portfolio is weighted average of Greeks of individual portfolio components

\[ \Delta_{\text{portfolio }} = \sum_{i=1}^{n} \omega_{i} \Delta_{i} \]

Implied Volatility

• Volatility is unobservable

• Option prices, particularly for near-the-money options, can be quite sensitive to volatility

• One approach is to compute historical volatility using the history of returns

• A problem with historical volatility is that expected future volatility can be different from historical volatility.

• Alternatively, we can calculate implied volatility, which is the volatility that, when put into a pricing formula (typically Black-Scholes), yields the observed option price.

Implied Volatility (cont’d)

• In practice implied volatilities of itm, atm, and out-of-the money options are generally different

• A volatility smile refers to when volatility is symmetric, with volatility lowest for at-themoney options, and high for in-the-money and out-of-the-money options

• A difference in volatilities between in-the-money and out-of-the-money options is referred to as a volatility skew

Implied Volatility (cont’d)

Appendix 12.A: The Standard Normal Distribution

The standard normal probability density function is given by

\[ \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^{2}} \]

The cumulative standard normal distribution function, evaluated at point \(x\), for example, tells us the probability that a number randomly drawn from the standard normal distribution will fall below \(x\), 1.

\[ N(x) = \int_{-\infty}^{x} \phi(x)dx \equiv \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^{2}} dx \]

The scipy module in Python computes this as: scipy.stats.norm.cdf

Appendix 12.B: Formulas for Option Greeks

Delta (\(\Delta\))

Delta measures the change in the option price for a \(\$1\) change in the spot price:

\[ \begin{aligned} \text{Call delta } &= \frac{\partial C(S, K, \sigma, r, T - t, \delta)}{\partial S} = e^{-\delta (T - t)} N(d_{1}) \\ \text{Put delta } &= \frac{\partial P(S, K, \sigma, r, T - t, \delta)}{\partial S} = -e^{-\delta (T - t)} N(-d_{1}) \end{aligned} \]

Appendix 12.B: Formulas for Option Greeks (cont’d)

Gamma (\(\Gamma\))

Gamma measures the change in delta when the spot price changes:

\[ \begin{aligned} \text{Call gamma } &= \frac{\partial^{2} C(S, K, \sigma, r, T - t, \delta)}{\partial S^{2}} = \frac{e^{-\delta (T - t)} N^{\prime}(d_{1})}{S\sigma\sqrt{(T - t)}} \\ \text{Put gamma } &= \text{Call gamma} \end{aligned} \]

The second equation follows from put-call parity.

Appendix 12.B: Formulas for Option Greeks (cont’d)

Theta (\(\Theta\))

Theta measures the change in the option price with respect to calendar time \((t)\), holding fixed time to expiration \((T)\) :

\[ \begin{aligned} \text{Call theta } &= \frac{\partial C(S, K, \sigma, r, T - t, \delta)}{\partial t} \\ &= \delta Se^{-\delta (T - t)} N(d_{1}) - rKe^{-r (T - t)} N(d_{2}) - \frac{Ke^{-r (T - t)} N^{\prime}(d_{2}) \sigma}{2 \sqrt{(T - t)}} \\ \text{Put theta } &= \frac{\partial P(S, K, \sigma, r, T - t, \delta)}{\partial t} \\ &= \text{Call theta } + rKe^{-r (T - t)} - \delta Se^{-\delta (T - t)} \end{aligned} \]

If time to expiration is measured in years, theta will be the annualized change in the option value. To obtain a per-day theta, divide by 365.

Appendix 12.B: Formulas for Option Greeks (cont’d)

Vega

Vega measures the change in the option price when volatility changes. Some writers also use the terms lambda or kappa to refer to this measure:

\[ \begin{aligned} \text{Call vega } &= \frac{\partial C(S, K, \sigma, r, T - t, \delta)}{\partial \sigma} = Se^{-\delta(T-t)} N^{\prime}(d_{1}) \sqrt{(T - t)} \\ \text{Put vega } &= \frac{\partial P(S, K, \sigma, r, T-t, \delta)}{\partial \sigma} = \text{ Call vega} \end{aligned} \]

It is common to report vega as the change in the option price per percentage point change in the volatility. This requires dividing the vega formula above by 100.

Appendix 12.B: Formulas for Option Greeks (cont’d)

Rho (\(\rho\))

Rho is the partial derivative of the option price with respect to the interest rate:

\[ \begin{aligned} \text{Call rho } &= \frac{\partial C(S, K, \sigma, r, T - t, \delta)}{\partial r} = (T - t) K e^{-r (T - t)} N(d_{2}) \\ \text{Put rho } &= \frac{\partial P(S, K, \sigma, r, T - t, \delta)}{\partial r} = -(T - t) K e^{-r (T - t)} N(-d_{2}) \end{aligned} \]

These expressions for rho assume a change in \(r\) of 1.0. We are typically interested in evaluating the effect of a change of 0.01 (100 basis points) or 0.0001 (1 basis point). To report rho as a change per percentage point in the interest rate, divide this measure by 100. To interpret it as a change per basis point, divide by 10,000.

Appendix 12.B: Formulas for Option Greeks (cont’d)

Psi (\(\psi\))

Psi is the partial derivative of the option price with respect to the continuous dividend yield:

\[ \begin{aligned} \text{Call psi } &= \frac{\partial C(S, K, \sigma, r, T - t, \delta)}{\partial \delta} = -(T - t) S e^{-\delta (T - t)} N(d_{1}) \\ \text{Put psi } &= \frac{\partial P(S, K, \sigma, r, T - t, \delta)}{\partial \delta} = (T - t) S e^{-\delta (T - t)} N(-d_{1}) \end{aligned} \]

To interpret psi as a price change per percentage point change in the dividend yield, divide by 100.